Optical (spectral) data from downhole fluid analysis logging tools are currently being used to determine the composition of crude oils downhole. See, e.g., Fujisawa, G., et al., “Near-infrared Compositional Analysis of Gas and Condensate Reservoir Fluids at Elevated Pressures and Temperatures,” Applied Spectroscopy, 52(12: 1615-1620 (2002); Fujisawa, G. et al., “Analyzing Reservoir Fluid Composition In-Situ in Real Time: Case Study in a Carbonate Reservoir”, SPE 84092, Annual Technical Conference and Exhibition, Denver, Colo. (2002) which are both hereby incorporated by reference herein in their entireties. These determinations are restricted to a subset of components or pseudocomponents including C1 (methane) to C5 (pentane), such as C1, C2-C5, and also C6+. The optical tools measure the optical densities {OD} at a set of wavelengths λi. These are then used to determine the weight percent wcj of components and pseudocomponents such as C1, C2-C5 and C6+, or C1, C2, C3-C5 and C6+. The individual weight percents for C2, C3, C4, and C5 can then be further resolved using a delumping algorithm. For some of the optical tools, the amount of water and carbon-dioxide (CO2) can also be determined. In addition, the optical density can be used to obtain information about asphaltenes and resins. See, e.g., Mullins, O. C., et al., “The Colloidal Structure of Crude Oils and the Structure of Reservoirs,” Energy Fuels, 21:2785-2794 (2007) which is hereby incorporated by reference herein in its entirety.
NMR relaxation and diffusion data can also be used to determine oil composition. From this data, the average chain length and the chain length distribution can be obtained. See, e.g., Freed, D. E., et al., “Scaling Laws for Diffusion Coefficients in Mixtures of Alkanes,” Phys. Rev. Lett., 94:067602 (2005); Freed, D. E., “Dependence on Chain Length of NMR Relaxation Times in Mixtures of Alkanes,” J. Chem. Phys., 126:174502 (2007); Hurlimann, M. D. et al., “Hydrocarbon Composition from NMR Diffusion and Relaxation Data,” SPWLA, 49th Annual Logging Symposium (May 2008); U.S. Pat. No. 6,859,032 to Heaton, N.J., and Freedman, R.; Anand, V. and Freedman, R., “New Methods for Predicting Properties of Live Oils from NMR,” SPWLA, Paper AAAA Proceedings of the 2009 Annual SPWLA Symposium (2009), which are all incorporated by reference herein in their entireties. In addition the comparison of transverse and longitudinal relaxation times and/or diffusion can give some information about asphaltenes, and the shapes of the distributions can be a signal of highly biodegraded oils. See, e.g., Mutina, A. R., and Hurlimann, M. D., “Correlation of Transverse and Rotational Diffusion Coefficient: A Probe of Chemical Composition in Hydrocarbon Oils,” J. Phys. Chem., A 112:3291-3301 (2008); Freed, D. E., and Hurlimann, M. D., “One- and Two-Dimensional Spin Correlation of Complex Fluids and the Relation to Fluid Composition,” C. R. Phys., 11:181-191 (2010), which are both hereby incorporated by reference herein in their entireties. Furthermore, the measurement of the NMR relaxation dispersion, i.e., the relaxation profile as a function of the applied magnetic field, can yield additional information about the aggregation propensity of the asphaltenes and resins in the crude oil.
NMR relaxation and diffusion measurements can be made with a downhole fluid analysis logging tool. See, Kleinberg, R. L., “Well logging”, Encyclopedia of Nuclear Magnetic Resonance, John Wiley (1996), which is hereby incorporated by reference herein in its entirety. The NMR tools measure the magnetization Mi at a series of echo times ti. They can also measure the magnetization as a function of wait times τi or as a function of bi, which is a diffusion weighting parameter determined by gradients and time variables. The tool data, such as {Mi, ti}, {Mi, τi}, or {Mi, bi} are used to determine the transverse or longitudinal relaxation time distributions or the diffusion distributions, given by {fj,T2j}, {fj,T1j}, or {fj,Dj}, respectively. For these distributions, fj is the fraction of protons with relaxation time T2j or T1j or with diffusion coefficient Dj, weighted by the total magnetization, M0. These distributions can be related to the raw data by an inversion process, such as an inverse Laplace transform. See, Fordham, E. J. et al., “Imaging multiexponential relaxation in the (y, logdT1) plane, with application to clay filtration rock cores,” J. Magn, Reson, A, 113:139-150 (1995); Venkataramanan, L. et al., “Solving fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions,” IEEE Trans. Signal Process, 50:1017-1026 (2002), which are both hereby incorporated by reference herein in their entireties. However, this inverse Laplace transform can be problematic because when noise is present, the inversion is non-unique. As a result, a regulator is often introduced to ensure the smoothness of the calculated distributions. These issues introduce some uncertainty into the calculated relaxation and diffusion distributions. See, Epstein, C. L., and Schotland, J., “The bad truth about Laplace's transform,” SIAM Rev., 50:504-520 (2008), which is hereby incorporated by reference herein in its entirety.
To obtain the relaxation or diffusion distribution from the raw data, the quantity ∥d−Kf∥2 is minimized with the constraint that fj, the components of the vector f, are non-negative. In this expression, d is the vector with components di=Mi, and κ is the kernel. For standard measurements, it is given by Kij=exp(ti/T2j), 1-2exp(−ti/T1j) and exp(−biDj) for the transverse relaxation, longitudinal relaxation and diffusion, respectively. In the past, the above expression is minimized using methods such as a non-negative least square fit with Tikhonov regularization or by maximum entropy methods.
Once the relaxation or diffusion distributions are known, the NMR data can be used to obtain information about chain length distributions and the viscosity of the oil. The viscosity η of the oil is related to the −1st moment of the diffusion coefficient. See, Freed, D. E., et al., “Scaling laws for diffusion coefficients in mixtures of alkanes,” Phys Rev Lett. 94:067602 (2005) and Hurlimann, M. D. et al., “Hydrocarbon composition from NMR diffusion and relaxation data,” SPLWA, 49th Annual Logging Symposium (May 2008). In terms of the log distribution, fD (log Di), the viscosity is given by
                              η          =                      CT            ⁢                                          ∫                                                                                                    f                        D                                            ⁡                                              (                                                  log                          ⁢                                                                                                          ⁢                                                      D                            i                                                                          )                                                              /                                          D                      i                                                        ⁢                                      ⅆ                                          log                      ⁡                                              (                                                  D                          i                                                )                                                                                                                        ∫                                                                            f                      D                                        ⁡                                          (                                              log                        ⁢                                                                                                  ⁢                                                  D                          i                                                                    )                                                        ⁢                                      ⅆ                                          log                      ⁡                                              (                                                  D                          i                                                )                                                                                                                                ,                            (        1        )            where the temperature T is in degrees Kelvin and C has been found to be 3.2×10−8cpcn2sK, but can vary somewhat depending on the type of oil. For numerical calculations, the integrals in Eq. (1) are replaced with summations. There are also correlations between viscosity and the mean log diffusion coefficient or relaxation time which appear in the literature. See, e.g., Morriss, C. E., et al., “Hydrocarbon saturation and viscosity estimation from NMR logging in the Beldridge diatomite,” The Log Analyst, 382:44-59 (1996); Kleinberg, R. L., and Vinegar, H. J., “NMR properties or reservoir fluids,” The Log Analyst, 376:20-32 (1996); Lo, S., et al., “Correlations of NMR relaxation time with viscosity, diffusivity, and gasoil ratio of methanehydrocarbon mixtures,” Proceedings of the 2000 Annual Technical Conference and Exhibition, Society of Petroleum Engineers (October, 2000); Straley, C., “Reassessment of correlations between viscosity and NMR measurements,” SPWLA, 47th Annual Logging Symposium (June 2006) which are all hereby incorporated by reference herein in their entireties.
Several methods have been proposed to relate NMR relaxation and diffusion to chain length distributions. One method makes use of radial basis functions to interpolate between known data and new measurements. See, Anand, V., and Freedman, R., “New methods for predicting properties of live oils from NMR,” Paper AAAA Proceedings of the 2009 Annual SPWLA Symposium (2009). Another method uses the constituent viscosity model to relate the diffusion coefficients and relaxation times of each component to its microscopic, or constituent, viscosity. See, previously incorporated U.S. Pat. No. 6,859,032. A third method as discussed below is based on looking at mixtures of alkanes, but can apply to oils with other components also.
For the method based on looking at mixtures of alkanes, the average chain length or carbon number (the terms “chain length” and “carbon number” being used interchangeably herein) is defined as N=ΣxjNj, where xj is the mole percent of molecules with chain length Nj. For oils high in saturates, this average chain length is related to the 1/vth moment of the diffusion distribution and, in the absence of asphaltene, to the 1/κth moment of the relaxation time distribution, where v=0.7, and κ=1.24. They are given by Freed, D. E., et al., “Scaling laws for diffusion coefficients in mixtures of alkanes,” Phys Rev Lett. 94:067602 (2005), and Freed, D. E., “Dependence on chain length of NMR relaxation times in mixtures of alkanes,” J. Chem, Phys., 126:174502 (2007):
                                          N            _                    =                                    A                              1                                  β                  +                  v                                                      ⁢                                          〈                                  D                                      1                    /                    v                                                  〉                                                              -                  v                                                  β                  +                  v                                                                    ,        and                            (        2        )                                          N          _                =                              B                          1                              γ                +                κ                                              ⁢                                                    〈                                  T                                      1                    ,                    2                                                        1                    /                    κ                                                  〉                                            -                                  κ                                      γ                    +                    κ                                                                        .                                              (        3        )            In these equations, A and B are constants that depend on temperature and pressure, and β and γ are constants that depend on temperature. The chain length Ni that corresponds to the diffusion coefficient Di is then given by previously incorporated Freed, D. E., et al., “Scaling laws for diffusion coefficients in mixtures of alkanes,” Phys Rev Lett. 94:067602 (2005),Ni=A1/vN−β/vDi−1/v  (4)
For chain lengths less than about five, this expression should be modified, because, in that case, the molecules act more like hard spheres than chains. Similarly, in the absence of asphaltenes, the chain length that corresponds to the relaxation time is given2i by previously incorporated Freed, D. E., “Dependence on chain length of NMR relaxation times in mixtures of alkanes,” J. Chem, Phys., 126:174502 (2007),Ni=B1/κN−γ/κT1,2i−1/κ  (5)
It should be appreciated that equation (5) is not valid for dissolved gases, such as methane and ethane, because they relax by different processes than the longer molecules. If the diffusion or relaxation distribution was determined as function of log Di or log log T1,2i with the log Di or log T1,2i evenly spaced, then the log distribution for the proton fraction of spins on molecules with chain length Ni is given byfN(log Ni)=vfD(log Di),  (6)fN(log Ni)=κfT(log T1,2i),  (7)respectively. The weight fraction, on a log scale, is then given by
                              w          ⁡                      (                          log              ⁢                                                          ⁢                              N                i                                      )                          =                                                            [                                                      (                                                                  7                        ⁢                                                  N                          i                                                                    +                      1                                        )                                    /                                      (                                                                  N                        i                                            +                      1                                        )                                                  ]                            ⁢                                                f                  N                                ⁡                                  (                                      log                    ⁢                                                                                  ⁢                                          N                      i                                                        )                                                                    ∫                                                [                                                            (                                                                        7                          ⁢                                                      N                            j                                                                          +                        1                                            )                                        /                                          (                                                                        N                          j                                                +                        1                                            )                                                        ]                                ⁢                                                      f                    N                                    ⁡                                      (                                          log                      ⁢                                                                                          ⁢                                              N                        j                                                              )                                                  ⁢                                  ⅆ                                      log                    ⁡                                          (                                              N                        j                                            )                                                                                                    .                                    (        8        )            In this equation, it is assumed that a molecule with N, carbon atoms has 2Ni+2 protons. Then, on a linear scale, the weight fraction of molecules with chain length Ni is given bywi=w(log Ni)/Ni  (9)
It should be appreciated that equations (2) through (5) set forth above for chain length and mean chain length as a function of diffusion coefficients and relaxation times may be derived from the observation that, for oils high in saturates, the diffusion coefficient have the formDi=AN−βNi−v  (10)and similarly for relaxation times. For small molecules such as methane and ethane, the quantity Niv in Eq. (10) is modified because the molecules no longer act like chain molecules. For methane, it is replaced with 1.64, and for ethane, it is replaced with 2.73. See, Freed, D. E., et al., “Scaling laws for diffusion coefficients in mixtures of alkanes,” Phys Rev Lett. 94:067602 (2005). In this way, the NMR relaxation and diffusion distributions can give the chain length distribution for the entire oil, not just for components below C6. However, the resolution is not particularly good.